0132_0213_0231_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 134, 298, 613, 1198, 2272, 4250, 7935, 14902, 28266, 54218, 105097, 205534, 404740, 801082, 1591235, 3168486, 6319342, 12616746, 25206509, 50380174, 100720744, 201394138, 402732103, 805398038, 1610718642, 3221347210, 6442590225, 12885060542, 25769983756, 51539810938, 103079444107, 206158687174, 412317147830, 824634041322, 1649267798005, 3298535278446, 6597070203632, 13194140015386, 26388279597199, 52776558715894, 105553116904954, 211106233231178, 422212465827993, 844424930962078, 1688849861166612, 3377699721507770, 6755399442117715, 13510798883260582, 27021597765464446, 54043195529785258, 108086391058334717, 216172782115336014, 432345564229235320, 864691128456924762, 1729382256912188375, 3458764513822594006, 6917529027643277122, 13835058055284508426, 27670116110566829089, 55340232221131321214, 110680464442260148764, 221360928884517639418, 442721857769032448283, 885443715538061885318, 1770887431076120570182, 3541774862152237741930, 7083549724304471878405, 14167099448608939935022, 28334198897217875822336, 56668397794435747361178, 113336795588871490192927, 226673591177742975600054, 453347182355485946147210, 906694364710971886963402, 1813388729421943768306345, 3626777458843887530691166, 7253554917687775055147812, 14507109835375550103735866, 29014219670751100200574179, 58028439341502200393900134, 116056878683004400780188174, 232113757366008801552386858, 464227514732017603096392973, 928455029464035206183999758, 1856910058928070412358793352, 3713820117856140824707945690, 7427640235712281649405800295, 14855280471424563298801043862, 29710560942849126597591049426, 59421121885698253195170562698, 118842243771396506390329074737, 237684487542793012780645567294, 475368975085586025561278003500, 950737950171172051122542309242, 1901475900342344102245070335915
Generating function in Maple syntax:
-(3*x^6+5*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{3 x^{6}+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-3*x**6 - 5*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+3*x^6+5*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+3 x^{6}+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1+17/6*n^2-29/6*n-7/6*n^3+1/6*n^4+3*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1+\frac{17 n^{2}}{6}-\frac{29 n}{6}-\frac{7 n^{3}}{6}+\frac{n^{4}}{6}+3 \,2^{-1+n} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 134
a(n+1) = 2*a(n)-1/6*(-1+n)*(n^3-10*n^2+22*n-24), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 134
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-1+n \right) \left(n^{3}-10 n^{2}+22 n -24\right)}{6}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18212/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[12,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[9,x]
F[27,x] = F[14,x]
F[28,x] = F[20,x]
F[29,x] = F[30,x]+F[31,x]+F[50,x]
F[30,x] = 0
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[49,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[30,x]+F[35,x]+F[41,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[10,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[37,x]
F[40,x] = F[17,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[14,x]+F[44,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[43,x]
F[49,x] = F[29,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[60,x]
F[52,x] = F[20,x]+F[53,x]
F[53,x] = F[30,x]+F[54,x]+F[59,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[18,x]+F[30,x]+F[35,x]
F[58,x] = F[53,x]
F[59,x] = F[12,x]*F[52,x]
F[60,x] = F[61,x]+F[73,x]
F[61,x] = F[30,x]+F[62,x]+F[71,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[14,x]+F[65,x]
F[65,x] = F[30,x]+F[45,x]+F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[68,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[20,x]+F[61,x]
F[73,x] = 2*F[30,x]+F[74,x]+F[78,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[17,x]+F[77,x]
F[77,x] = F[48,x]
F[78,x] = F[12,x]*F[60,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{50}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{49}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{41}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{29}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{58}\! \left(x \right) = F_{53}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{73}\! \left(x \right)
F_{61}\! \left(x \right) = F_{30}\! \left(x \right)+F_{62}\! \left(x \right)+F_{71}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right)+F_{45}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{20}\! \left(x \right)+F_{61}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{17}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{48}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_19(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_9(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_20(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_50(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_49(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_30(x) + F_35(x) + F_41(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_10(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_37(x))
Eq(F_40(x), F_17(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_14(x) + F_44(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_43(x))
Eq(F_49(x), F_29(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_60(x))
Eq(F_52(x), F_20(x) + F_53(x))
Eq(F_53(x), F_30(x) + F_54(x) + F_59(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_18(x) + F_30(x) + F_35(x))
Eq(F_58(x), F_53(x))
Eq(F_59(x), F_12(x)*F_52(x))
Eq(F_60(x), F_61(x) + F_73(x))
Eq(F_61(x), F_30(x) + F_62(x) + F_71(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_14(x) + F_65(x))
Eq(F_65(x), F_30(x) + F_45(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_68(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_20(x) + F_61(x))
Eq(F_73(x), 2*F_30(x) + F_74(x) + F_78(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_17(x) + F_77(x))
Eq(F_77(x), F_48(x))
Eq(F_78(x), F_12(x)*F_60(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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